Riemann for Anti-Dummies Part 24
LET THERE BE LIGHT
As you heard Riemann proclaim in the opening remarks of his Habilitation lecture, without a “general concept of multiply-extended magnitudes in which spatial magnitudes are comprehended,” you are left in the dark. You can not know the nature of the physical universe, the validity of an idea, the economic value of human activity, the strategic significance of a current, or historical, event, or your personal identity in the simultaneity of eternity, to name but a few of the more important matters on which one would wish to shed light. Yet, the principles to which Riemann refers are far too little understood by those who must urgently be able to make such judgements.
Referencing Gauss, Riemann cites two characteristics necessary for the determination of multiply-extended magnitudes, dimensionality and curvature, neither of which can be determined a priori, but only by physical measurement. Such magnitudes are not mathematical quantities, but are universal physical principles, produced by a manifold of physical action, and, are relative to the manifold, not absolute.
Take some examples from the arsenal of ideas built up over the course of this series to illustrate the point.
1. As Kepler demonstrated, the non-uniform elliptical planetary orbit defines the magnitude of action within an orbit, as equal areas, rather than the arbitrary mathematical magnitudes of equal arcs or equal angles. The solar system as a whole, in turn defines a magnitude of action for individual orbits, consistent with the five Platonic solids, and the principles of musical polyphony. Thus, the action of a planet at any moment can only be measured as a function of the whole orbit, which orbit in turn is measured as a function of the whole solar system. While the orbit defines one species of magnitude (equal areas) the solar system as a whole defines a distinct and different species of magnitude (harmonics), which “reach down” into all parts of the individual orbit, even though the latter cannot be derived simply from the former.
2. The shortest path of reflected light defines a magnitude of action measured by equal angles. The least time path of refracted light defines a magnitude of action measured by the proportionality of the sines of the angles of incidence and refraction. In other words, under reflection the angles measure the change in the direction of the light, while under refraction, the angles are determined by the sines. In the manifold of physical action of reflected light, there is no change in medium, consequently no change in velocity of light, and so the effect of the sines “disappears” into the equality of angles. But in the higher dimensional manifold of refraction, the truth comes out, that it is not the angles that measure the action, but the inverse, the transcendental magnitudes of the sines.
It is important to keep in mind, that in both these examples, “dimension” is not a mathematical construct, but is associated with a distinct physical principle, which is then associated with a distinct species of magnitude, and, as Riemann emphasizes, the number of dimensions is increased by the discovery of each new physical principles.
This concept of magnitude is consistent with Schiller’s expression in “On the Aesthetic Estimation of Magnitude”:
“All comparative estimation of magnitude, however, be it abstract or physical, be it wholly or only partly determined, leads only to relative, and never to absolute magnitude; for if an object actually exceeds the measure which we assume to be a maximum, it can still always be asked, by how many times the measure is exceeded. It is certainly a large thing in relation to its species, but yet not the largest possible, and once the constraint is exceeded, it can be exceeded again and again, into infinity. Now, however, we are seeking absolute magnitude, for this alone can contain in itself the basis of a higher order, since all relative magnitudes, as such, are like to one another. Since nothing can compel our mind to halt its business, it must be the mind’s power of imagination which sets a limit for that activity. In other words, the estimation of magnitude must cease to be logical, it must be achieved aesthetically.
“If I estimate a magnitude in a logical fashion, I always relate it to my cognitive faculty; if I estimate it aesthetically, I relate it to my faculty of sensibility. In the first case, I experience something about the object, in the second case, on the contrary. I only experience something within me, caused by the imagined magnitude of the object. In the first case I behold something outside myself, in the second, something within me. Thus, in reality, I am no longer measuring, I am no longer estimating magnitude, rather I myself become for the moment a magnitude to myself, and indeed an infinite one. That object which causes me to be an infinite magnitude to myself, is called sublime.”
Think in these terms about Gauss’ development of the complex domain in the context of his work on biquadratic residues, where Gauss demonstrates that it is actually impossible to construct a concept of magnitude devoid of dimensionality. As the discoveries that Plato made famous in his Meno, Theatetus, and Timaeus dialogues, action along a line, a surface, or a solid, is associated, in each case, with distinct species of magnitude. The species of magnitude, associated with the manifolds of lower dimensions, are found in the manifolds of higher dimensions but not vice versa. Consequently, a paradox arises, if one attempts to measure action in manifolds of higher dimensions, by magnitudes that are produced in a manifold of lower dimensionality.
Look at this from the standpoint of the simple operations with numbers, addition, subtraction, multiplication, division. (Riemann added Leibniz’ integration and differentiation, to the domain of simple operations, and this will be taken up in future installments.) As the Theatetus reports, addition of doubly-extended magnitudes, (i.e. areas) cannot be measured by simply-extended magnitudes (i.e. lines), and yet, until Gauss, all operations of Arithmetic were constrained by the underlying assumption that each manifold could be measured by the same species of magnitudes. This paradox reemerged from the Renaissance on, as the paradox associated with the ?-1. Cardan, Leibniz, Huygens, and Kaestner, all understood that this paradox required the need for a higher conception of magnitude, while Newton, Euler and others, dismissed this magnitude as “impossible” .
For Gauss, action in a doubly-extended manifold, could only be measured by doubly- extended magnitudes, which he called “complex numbers”. These numbers are determined by two actions, rotation and extension, or alternatively, simultaneous horizontal and vertical action, such as in the bubble of a carpenters level. (It is about time to replace the commonly used term “Cartesian coordinates” when referring to horizontal and vertical action, with the more historically and conceptually accurate, term, “Fermat coordinates”.)
From this standpoint look at the basic concepts of Arithmetic with respect to both simply- extended and doubly-extended magnitudes. Under Gauss’ concept of congruence, all numbers are ordered with respect to the interval between them or the modulus. With respect to simply- extended manifolds, that interval corresponds to a line segment. But, with respect to a doubly- extended manifold, that interval has two parts, up-down and back forth. Illustrate this with an example from Gauss’ second treatise on bi-quadratic residues. The modulus 5+2i “partitions” the entire complex domain, by a series of squares whose sides are the hypothenuses of right triangles whose legs are 5 and 2. For example, the square whose vertices are 0, 5+2i, 3+7i, – 2+5i. All complex numbers inside this square are not congruent to each other. Now draw adjacent squares, such as the squares whose vertices are 5+2i, 10+4i, 8+9i, 3+7i; or, 3+7i, 8+9i, 6+14i, 1+12i. All the numbers inside these squares are also not congruent to each other, but each number is congruent to the one number in each of the other squares, which is in the same relative position within the square. For example, 2+4i and 7+6i, occupy the same relative position within their respective squares, consequently, the difference (interval) between them is the modulus 5+2i.
Thus, the simple periodicity generated by congruences with respect to real numbers, is transformed into a double periodicity with respect to complex moduli. In a simply-extended manifold, therefore, subtraction determines the linear interval between two numbers, while in a doubly-extended manifold, subtraction determines the area interval between two numbers.
Gauss next developed a concept of doubly-extended “complex” multiplication, which will require you to re-think what you were taught about multiplying numbers in elementary school. Simply-extended multiplication was defined by Euclid as:
“15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.”
But, even Euclid admits an inadequacy of this concept in the next definition:
“16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.”
But, we already have discovered from Theatetus, that adding in a simply-extended manifold, (lines) and adding in a doubly-extended manifold (areas) are not the same, so how can adding one number to itself “as many times as there are units in the other” (definition 15.) produce the areas described in definition 16? This darkness arises from the lack of a concept of multiply-extended magnitude.
The matter is resolved by a higher concept. If you look again at Theatetus’ alternating series of squares and rectangles, or the expanding series of squares from the Meno, you can see that adding areas produces a rotation and an extension. For example, the square whose area is 1 is transformed into the rectangle whose area is two, by rotating a line whose length is 1 90 degrees and multiplying its length by 2. The next transformation, to a square whose area is 4, is produced by the action of rotating the longer side of the rectangle and additional 90 degrees, and multiplying its length again by 2.
As Gauss’ follower Neils Henrick Abel said, “To know the truth, you must always invert.” An inversion, therefore, will show us the general principle that is, the action of adding rotation and multiplying lengths, produces the geometric progression.
So in the complex domain, multiplication is the action of adding rotations and multiplying lengths.
Illustrate this first with respect to prime numbers, by the example of multiplying (1+2i)(1-2i). 1+2i denotes a rotation of 45 degrees and a linear extension of ?5. 1-2i denotes a rotation of 45 degrees, in the opposite direction and a linear extension of ?5. To multiply the two magnitudes, add the rotations, (which together equal 0) and multiply the extensions (?5)(?5) = 5. Hence, 5, a prime number in a simply-extended manifold, is a composite number in a doubly- extended manifold. However, no such geometric action will produce odd-odd prime numbers such as 7, 11, 19, etc.
Gauss saw this paradox as an excellent pedagogical demonstration of the principle that the nature of the manifold determines nature of the magnitudes. Since prime numbers produce all numbers by multiplication, but cannot be produced themselves by multiplication, Gauss has shown that these magnitudes, (prime numbers) that produce other magnitudes (composite numbers) are themselves produced by the manifold in which the action (multiplication) takes place . Some numbers are prime (undeniable facts) but when a new principle (dimension) is added even those undeniable facts, are changed!
Now, construct a geometric progression from a complex number, by multiplying that number by itself repeatedly. For example, start with 1+i which denotes a rotation of 45 degrees and an extension of ?2. Then multiply 1+i times 1+i. This produces a rotation of 90 degrees and an extension of 2. Repeating this again produces a rotation of 135 degrees and an extension of 4. If you continue this action you will see unfolding points on a logarithmic spiral.
From this Gauss demonstrated that the periodicity produced by the residues of a geometric progression, actually reflected magnitudes of a higher manifold. In the next installment, we will illustrate this discovery.